The FCC Commissioner is appointed, per 47 USC 154, by POTUS, subject to approval by the US Senate (as a member of the Commission). Thereafter he serves for 5 years. However Article 2 of the Constitution allows removal from office:
The President, Vice President, and all civil Officers of the United
States shall be removed from Office on Impeachment for, and conviction
of, Treason, Bribery, or other High Crimes and Misdemeanors.
But only the House of Representatives can impeach and only the Senate can remove (of the President declines to do so). There is no provision for a citizen to sue to remove a civil officer. It is, of course, legal for citizens and others to lobby politically for whatever action that want Congress to take.
"Net Neutrality" rules can be more permanently fixed by act of Congress. By act of Congress, the FCC may be required to, allowed to, or prohibited from setting some set of rules. Such laws are typically broad and leave a lot of discretion to the executive branch. Congress has not passed a law mandating "net neutrality", and as ruled in Comcast v. FCC the FCC lacks "ancillary jurisdiction" over internet services of at least Comcast.
There have been previous attempts to legislate "net neutrality", such as the Internet Freedom and Nondiscrimination Act of 2006 which died on the floor of the House, which purported to guarantee "net neutrality" via an extension of antitrust law. However, the only way to make such a provision permanent is to make it be a Constitutional Amendment, since an act of Congress can be repealed or amended in such a way that it effectively doesn't exist.